Heterogeneous
Heterogeneous nucleation is the accumulation of seed nuclei in a liquid or gaseous environment, resulting from the presence of impurities of a different phase. This is why, in a solution, the presence of a substrate, such as a wooden dowel or flat surface of metal or rock, encourages the growing of crystals. It's much more likely to be spontaneous than a homogeneous nucleation.
Homogeneous
Homogeneous nucleation is encouraged by superheating or supercooling a totally pure solution to create a "bubble" reaction or formation. The lack of any foreign substance, particulate matter or bounding walls allows a total reaction that changes the entire nature of the nucleated substance.
Phase Change
The entire aim of nucleation is phase change, from gas to liquid, from liquid to solid, and vice versa, in the case of homogeneous reactions. The process of condensation and precipitation are products of spontaneous nucleation, aggravated by some external force such as temperature or gravity.
Applications
The carbonation in drinks is an artificially induced nucleation reaction, created by the release of pressure of opening a bottle. The principle of nucleation is used to derive chemical concoctions that seed clouds and induce precipitation. Finally, many polymers are created in this way, phase changed from liquid to solid by heterogeneous reactions.